- Involution (mathematics)
In

mathematics , an**involution**, or an**involutary function**, is a function that is its own inverse, so that:"f"("f"("x")) = "x" for all "x" in the domain of "f".

**General properties**Any involution is a

bijection .The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include

multiplication by −1 inarithmetic , the taking of reciprocals, complementation inset theory and complex conjugation.Other examples include

circle inversion , theROT13 transformation, and the Beaufortpolyalphabetic cipher .**Involutions in Euclidean geometry**A simple example of an involution of the three-dimensional

Euclidean space is reflection against a plane. Performing a reflection twice brings us back where we started.This transformation is a particular case of an

affine involution .**Involutions in linear algebra**In linear algebra, an involution is a linear operator "T" such that $T^2=I$. Except for in characteristic 2, such operators are diagonalizable with 1's and -1's on the diagonal. If the operator is orthogonal (an

**orthogonal involution**), it is orthonormally diagonalizable.Involutions are related to idempotents; if 2 is invertible, (in a field of characteristic other than 2), then they are equivalent.

**Involutions in ring theory**In

ring theory , the word "involution" is customarily taken to mean anantihomomorphism that is its own inverse function.Examples of involutions in common rings:

*complex conjugation on thecomplex plane

* multiplication by j in thesplit-complex number s

* taking thetranspose in a matrix ring"See also

star-algebra ."**Involutions in group theory**In

group theory , an element of a group is an involution if it has order 2; i.e. an involution is an element "a" such that "a" ≠ "e" and "a"^{2}= "e", where "e" is theidentity element . Originally, this definition differed not at all from the first definition above, since members of groups were always bijections from a set into itself, i.e., "group" was taken to mean "permutation group ". By the end of the 19th century, "group" was defined more broadly, and accordingly so was "involution". The group of bijections generated by an involution through composition, is isomorphic withcyclic group "C"_{2}.A

permutation is an involution precisely if it can be written as a product of one or more non-overlapping transpositions.The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the

classification of finite simple groups .Coxeter group s are groups generated by involutions with the relations determined only by relations given for pairs of the generating involutions. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions.**Involutions in mathematical logic**The operation of complement in Boolean algebras is an involution. Accordingly,

negation in classical logic satisfies the "law of double negation:" ¬¬"A" is equivalent to "A".Generally in non-classical logics, negation which satisfies the law of double negation is called "involutive." In algebraic semantics, such a negation is realized as an involution on the algebra of truth values. Examples of logics which have involutive negation are, e.g., Kleene and Bochvar

three-valued logic s, Łukasiewicz many-valued logic,fuzzy logic IMTL, etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual e.g. int-norm fuzzy logics .The involutiveness of negation is an important characterization property for logics and the corresponding varieties of algebras. For instance, involutive negation characterizes Boolean algebras among

Heyting algebra s. Correspondingly, classical Boolean logic arises by adding the law of double negation tointuitionistic logic . The same relationship holds also betweenMV-algebra s andBL-algebra s (and so correspondingly betweenŁukasiewicz logic and fuzzy logic BL), IMTL and MTL, and other pairs of important varieties of algebras (resp. corresponding logics).**Count of involutions**The number of involutions on a set with "n" = 0, 1, 2, ... elements is given by the

recurrence relation ::"a"(0) = "a"(1) = 1;:"a"("n") = "a"("n" − 1) + ("n" − 1) × "a"("n" − 2), for "n" > 1.The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 OEIS|id=A000085.

**See also***

Automorphism

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